Place Value...a teaser

EDCURRIC 714 Exploring Mathematical Thinking taught by Gail Ledger at the University of Auckland is a great paper for learning about how humanity has thought about how number systems work and how the place value system came about. The practical values come in understanding where to start developing the concept of "one" or a single unit and building that into the concept of place value among other mathematical concepts.

Follow this link to the University of Auckland calendar website.

As it happens, the next weak spot in my groups' understanding to work on is place value. I used the idea of teaching the Ancient Egyptian numbers and numeral system last year to teach place value. The kids enjoyed it. Whether it helped them further develop an understanding of place value, I don't know. This coming term I'll be doing something similar, but a bit more 'hands on' and a bit more localised...

Ancient Egyptian numbers. Source: Grauberg, Eva (see below).

Now, time for the obligatory declaring of evidence that I have done some reading on the topic and not just pulled these ideas out of thin air. There are many texts on place value and ancient number systems, but here are a few (no, I'm not using any particular format e.g. APA).

Place Value:

• Elementary mathematics and language difficulties; a book for teachers, therapists and parents (1998). Grauberg, Eva.
• Children's Developing Understanding of Place Value: Semiotic Aspects. Maria Varelas and Joe Becker. Cognition and Instruction, Vol. 15, No. 2 (1997), pp. 265-286

Number history:

• Numbers through the ages (1989), Written Numbers pp. 76--130. Flegg, Graham.
• The Beginnings of Arithmetic, The Mathematical Gazette, Vol. 12, No. 177 (Jul., 1925), pp. 401-414. L. N. G. Filon

Ancient number systems:

• Ancient Egyptian - see above sources
• Chinese -
• Joseph, G. (2011). The crest of the peacock: non-European roots of mathematics (3rd ed.). Princeton: Princeton University Press
• Pellatt, V. (2007). Numbers and numeracy in Chinese culture, language, and education : the social substratum of the development of mathematical thinking.

IKAN results... a small ray of hope

My pupils were tested on their knowledge earlier this term. Generally in New Zealand, this involves the IKAN test. My pupils however often need to be retested on the JAM (junior assessment for maths) to get a clearer idea of what they know as opposed to what they can hear, read, parse, recall, then write down within 3 seconds.

After seeing the results, I had a sense of a general increase in certain domains of maths, especially fractions. I have put it to the test and made a comparison.

Colour coded IKAN shift for 2018

With green being the code for an increase in level, this screenshot of my spreadsheet shows that there has indeed been a general improvement in the Overall Level out of the pupils present for both tests (there is one coding error for pupil #28, which made no overall change). IKAN's Overall Level though, is taken from the lowest score within the domain scores, so it doesn't necessarily show improvement. The large number of increases in fractions demonstrates this phenomenon.

I have taken the small win in fractions and have moved on to improving basic facts for the remainder of this term (since about Week 7), and next term I will focus on place value.

Basic Facts Boxes

Basic Facts... a gap in knowledge that I never had when I was growing up. A gap in knowledge that prevents or hinders improvement in most areas of mathematics. How can Basic Facts knowledge in children, who are in some cases quite far behind expected ability, be brought back to average and boosted further?

Here's something I developed to give it a go.


This Google Sheet populates boxes with random addition and subtraction problems to solve. On entering an answer, the sheet checks the answer and gives immediate feedback. Another sheet in this Spreadsheet does the same for times tables and the corresponding division facts.

I tested the first version of this on my own pupils to limited success. The positive is that many of the kids wanted to keep going until they got everything correct. The negative was that they didn't take screenshots of the boards that had errors, so there was no record of their improvement.

A colleague, Robyn Anderson tested version 2 (without the times tables) with her class. You can read about her success and outcomes I never expected from this tool on her blog post.

The inspiration for this tool came from two places. The first being a Japanese tool for memorising basic facts and the second being Monty Jones's (Tāmaki College) spreadsheets for practising maths problems at secondary school level.

Feel free to copy it, use it, attribute it, and tell me about the successes and challenges you have had.

Maths Vlog

In another attempt to encourage learners into speaking about maths and develop their understanding of maths, I am trialling a task I'm calling Maths Vlog...again until I can think of a better name. This task involves learners creating a stack of objects in front of them and talking about the objects in relation to the concept being learned. This monologue is recorded then embedded into a Google Slide as if creating a vlog.

As can be seen in the example, the task is reusable (unlike worksheets), realia based (unlike many apps), and has some basic scaffolding to aid in learning.

The idea of Maths Vlog came from the success of SSR Selfie for promoting reading and FlipGrid for talking about maths. Hopefully this will be as successful as SSR Selfie in engaging learners with their basic maths concepts and provided the much needed mileage in using maths verbally.

Less than? Before? Minus one? What does that mean?

Observation of learners over the past years have shown that children in the tail end often have severe limitations in their vocabulary and understanding of basic concepts in size, order, and direction among others. Concepts need to be taught and repeated before internalisation occurs and application to problems can happen. How can I do this while meeting the demands of a full schedule? I know about apps, but frankly many are naff or incomplete for my needs. The many resources available from teachers around the world are often too babyish for my year 4/5/6 learners because these concepts should be developed in kindergarten or the first couple of years at primary school.

My solution is a series of semi open-ended tasks dubbed "Knowledge Cards"...until I can think of a cooler name. These are used with maths materials or other realia around the classroom to reinforce concepts in the learner's mind. Learners are also supposed to describe what they have done aloud in a Maths Vlog to aid in both reinforcing concepts and learning how to use the vocabulary verbally.

So far there has been a positive response to the cards I have developed. There are still some cards that I would like to make for both simple and more advanced maths concepts such as dividing/sharing items and part-whole number relationships.

Talking about capacity using Flipgrid

Capacity. What is it? How can it be measured? How is it usually measured? These are some questions for a maths concept that can develop some much needed language skills. But this is maths, how does one get children talking about maths? FlipGrid to the rescue.

In these lessons about capacity, children learned about containers and different ways of filling them up to find out how much they held. Children then learned how to talk about the comparative sizes of containers and how to measure exact capacities using a measuring cylinder and millilitres.

The capacity and verbal components to this lesson were very successful. The learners enjoyed using water and scientific instruments to find out about capacity. They also enjoyed sharing their learning by talking about it and using a simple video response recording system.

Problems arose with the word problems I had developed for them based on the learning done in the lesson. There were two main problems, both of which could be foreseen with knowledge of my learners. These problems were the inability to comprehend the problem despite being written in a simple and consistent format, and a lack of basic facts knowledge.

In subsequent lessons I structured and walked through how to understand, respond, and answer these problems with some success. I will need to continue scaffolding responses till learners become reasonably proficient before dropping the scaffold.

 

Concept Star - speak, please!

How do I promote speaking and sharing ideas to my maths learners? I had no idea. I have hunches that there are confidence issues as well as oral language issues as I have discussed before. The ideas mentioned revolved around ESOL and underdeveloped oral language skills, however I also noticed that my learners for reading and writing (whom shouldn't have many issues with communication) were willing to share ideas within small groups, but not as a class.

After trying a talking hands approach and gaining some traction, I decided to try using a Concept Star with my reading groups as a testing ground before using it with my maths learners. Concept Stars are used by learners to note down concepts the have learned before telling others. See how it was used with two of my reading groups here:

Here is a Google Drawing template of the Concept Star adapted from the TKI ESOL Online page about speaking strategies. If you use this template, please leave a comment about how your lesson went.

Talking Hands, empowering learners to speak their minds.

Oral language is becoming an increasingly focused upon area of development where I teach in particular. Last week's PD was on a number of ways to promote speaking in the class. Several strategies were covered, most being reminders of strategies that I have been taught over my limited experience as a teacher or during teacher training.

One strategy which I had initially dismissed was talking hands. Showing learners hand signals they can use for common responses helps them develop the confidence to give their opinion without the pressure of having to say anything. Combining this with conversing with learners rather than questioning them helps teaching them how to have conversations, how to discuss differing opinions, and awareness of how their speech sounds to others.

Reflection on the needs of my target students, I decided to give both the conversation and talking hands strategies a try in class. I used both on my reading groups which yielded reasonable results for the most part, with one group developing the confidence to speak out to correct a peer's incorrect statement in a constructive manner. My maths learners were shown hand signals only, to which there was a good response. 

To develop the confidence to speak in my learners, I will need to continue with the talking hands and conversation style of teaching. Hopefully this will increase their speaking "mileage" and exposure to new vocabulary and ways of communicating my modelling.

SSR Selfie

About a month ago I heard something I'd never heard in a class before, "Yay, SSR!" This year (2018), I am teaching the learners in my class that generally enjoy reading and comprehend their reading better, however this child wasn't particularly into reading. I decided to investigate deeper and found that my learners from last year who weren't particularly into reading also enjoyed my SSR activity.

SSR Selfie is an independent reading activity that I tried to model on social media use without actually getting onto social media thus keeping a bit more control and safety for the learners. The activity involves learners reading a book, taking a selfie with their book, then retelling or explaining what they have read. For some learners this could be an entire book or text, for others it could be a few pages in a novel they're reading. The point is that they discuss what they have read during their SSR time. The activity doesn't stop there however. Because I have made it on a shared Google Slide, this gives other learners the opportunity to check out what their classmates or friends have been reading and leave a comment. The children enjoy being able to talk about what they have been reading and leaving comments on other children's posts.

What is the upshot? Integrating learner voice into a task can yield increased motivation and buy-in for a learning activity. I want to try something similar for writing and maths activities.

Below are some examples of how it worked out this term. Here is a link to the template. If you like it, please leave a comment.

Maths Synonyms - reflection

The synonyms task I tried earlier this week was not as successful as I'd hoped. I had not anticipated two things: the learners misunderstanding the task and the learners not looking up the meanings of words they were unfamiliar with.

I still feel the vocabulary task would have been a valuable task for the learners to engage. Learners would be exposed to both the connections between everyday and mathematical vocabulary, and the nuances between synonyms such as trim and cut.

Ultimately, a positive is that the learners gained a stronger understanding of the meaning of the words addition, subtraction, multiplication, and division. However these synonyms will most likely need to be revisited someday.

Maths Synonyms

Bolstered by my mild success from the basic vocabulary lesson, I decided to try connecting the language of mathematical operations to everyday vocabulary they may already know. The idea was to also expand their vocabulary with related words they may not have already encountered.

In the lesson, learners focused on the core mathematical operations add, subtract, multiply, and divide. In groups, they used beans to demonstrate and explain the actions made by the operations.

The follow up task required learners to make videos demonstrating the meanings of the following words: sum, take, split, count up, remove, withdraw, total, cut, reduce, group, trim, share, tally, decrease. The words and videos were to be organised under the headings of the main operations.

Find, Draw, Animate, Act - Maths version

After trying a lesson on the language of subtraction using word problems, I decided to get back to basics.

In this lesson I went back to understanding the vocabulary behind quantity, magnitude, position, and direction. In the lesson, learners looked at the difference between big and tall first, then after discussion, the difference between much and many. Listening to learner discussion, I confirmed that many holes in understanding some of the more basic concepts such as big/small, before/after, first/last.

The task to complete was a vocabulary tool I have used before in reading lessons called Find, Draw, Animate, Act. In this activity, learners complete different tasks to demonstrate the meaning of words they have just looked up. The task is normally an open task for learners to input any new words they have learned from a text, however in this case I deliberately input the maths vocabulary. Selecting the words and tasks the learners had to do for each word lowered the cognitive load, with the intention that learners focused on looking up the meanings of words.

Following are a copy of the learning slide and the list of vocabulary.

Subtraction language - reflection

The subtraction language task was a almost a dismal failure. Many of  my learners were not able to unpack the language in the task. I will have to step back to more of the core concepts and build up language from there. I will also need to ensure tasks are explained and scaffolded better.

Subtraction language

The first thing I tried for increasing vocabulary was looking at word problems and decoding/ unpacking the language of subtraction problems.

In the lesson, I used two simple word problems to unpack the language with learners. We worked through understanding what a given problem might look like physically by trying to comprehend the sentences.

For the follow up task (click here to view), I used a basic worksheet style task with mixed addition and subtraction problems. Learners had to cut and paste word problems into either the Addition table, or the Subtraction table depending on what they thought the problem was. Learners were then encouraged to solve the problems, but this was not set as a compulsory task because I wanted them to focus on understanding the problem rather than rushing in to try and solve it.

Target Students

My target students for this year are as follows:

What are their learning needs?

• Overall need for oral and numeracy language skills.
• Overall need to understand how situations translate into mathematical problems.
• Students A, B, and C require a lot of basic mathematical knowledge.
• Student D has some knowledge base, but difficulties understanding written problems.

What strategies and skills are most likely to help my students?

• Vocabulary lessons and activities.
• Hands on activities for:
• number sequence
• number bonds
• fractions
• counting in groups
• addition and subtraction strategies using materials

Oral language activities

This resource for oral language was introduced to me to help with my inquiry into improving maths achievement through vocabulary and oral language development.

There are many strategies for developing confidence and ability in oral communication as well as developing listening comprehension. The strategies have their base in ESOL language support, making them relevant for the learners I will be teaching this year. Many of the learners I will teach maths this year have under developed oral language abilities or are ESOL due to their parents being immigrants to New Zealand themselves.

http://esolonline.tki.org.nz/ESOL-Online/Planning-for-my-students-needs/Resources-for-planning/ESOL-teaching-strategies/Oral-Language

2018 Teacher Inquiry

This year I will be looking into lifting maths achievement in my group of year 4, 5, and 6 learners through vocabulary and oral language.

I have noticed for some time now, many children coming up into my class have gaps in core maths knowledge such as number sequence past 100; basic addition, subtraction, and multiplication facts; and sharing/division objects and sets of things. In particular, one of the most commonly occurring observations I have made is misunderstandings of key vocabulary/ concepts such as 'before', 'after', or 'share', and their relationship to the language of maths problems they may encounter when needing to explain a problem or reading a test. These observations have continued to the beginning couple of weeks of this year leading me to the hunch that working with vocabulary and core maths concepts will lift achievement for my group of learners.